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# Types of matrices

Definition.
A square matrix is a matrix in which the number of rows equals the number of columns (size n×n), the number n, is called the order of the matrix.
Example. 4 1 -7 - a square matrix with size 3×3 -1 0 2 4 6 7

Definition.
The zero matrix is a matrix whose entries are equal to zero, ie, aij = 0, ∀i, j.
Example. 0 0 0 - zero matrix 0 0 0

Definition.
Row vector is a matrix consisting of a one row.
Example. 1 4 -5 - row vector

Definition.
Column vector is a matrix consisting of a one column.
Example. 8 - column vector -7 3

Definition.
Diagonal matrix is a square matrix whose entries standing outside the main diagonal are equal zero.
Example of diagonal matrix. 4 0 0 - diagonal entries are arbitrarynot diagonal entries are equal to zero 0 5 0 0 0 0

Definition.
Identity matrix is a diagonal matrix in which all the elements on the main diagonal are equal to 1.
Denote.
The identity matrix usually denoted by I.
Example of identity matrix.
 I = 1 0 0 - diagonal entries are equal to 1not diagonal entries are equal to 0 0 1 0 0 0 1

Definition.
Upper triangular matrix is a matrix whose elements below the main diagonal are equal to zero.
Example of upper triangular matrix. 7 -6 0 0 1 6 0 0 0

Definition.
Lower triangular matrix is a matrix whose elements above the main diagonal are equal to zero.
Example of lower triangular matrix. 7 0 0 6 1 0 -2 0 5

N.B. The diagonal matrix is a matrix that is both upper triangular and lower triangular.

Definition.
Specifically, a matrix is in row echelon form if:
• all nonzero rows are above all zero rows;
• if the first non-zero element of a row is located in a column with the number i, and the next line is not zero, then the first non-zero element of the next line should be in the column with number greater than i.
Examples of row echelon form of matrix. 7 0 8 0 0 4 7 0 8 8 8 0 0 1 3 5 0 0 0 -3 5 0 0 0 0 0 0 0 0 0 0